Question

a) Define (using truth tables) the disjunction, conjunction, exclusive or, conditional, and bi-conditional of the propositions p and q.

b) What are the disjunction, conjunction, exclusive or, conditional, and biconditional of the propositions “I’ll go to the movies tonight” and “I’ll finish my discrete mathematics homework”?

Short answer

The disjunction \(p \vee q\) is false when both p and q are false.

The conjunction \(p \wedge q\)is true when both p and q are true.

The exclusion\(p \oplus q\)is true when exactly one of p and q is true.

The conditional statement \(p \oplus q\)is true when exactly one of p and q is true.

The bi-conditional \(p \leftrightarrow q\)is true when p and q have the same truth values.

Step-by-step solution

Introduction

A proposition is a statement or assertion that expresses a judgment or opinion.

Truth Table is a diagram in rows and columns depicting the truthiness or falsity of a proposition varing with that of its components.

Disjunction of propositions

Define the disjunction of the propositions\(p\,\,and\,\,q\) follows:

The disjunction of\(p\,\,and\,\,q\)can be denoted by\(p \vee q\), which can be read as “p or q”. The disjunction\(p \vee q\)is false when both p and q are false and is true in all other cases.

The truth table for the disjunction\(p \vee q\)is as follows:

\(p\,\,\)	\(\,\,q\)	\(p \vee q\)
True	True	True
True	False	True
False	True	True
False	False	False

From the above table, observe that\(p \vee q\)is false when both p and q are false.

Conjunction of propositions

Define the conjunction of the propositions\(p\,\,and\,\,q\) follows:

The conjunction of\(p\,\,and\,\,q\)can be denoted by\(p \wedge q\), which can be read as “p and q”. The conjunction\(p \wedge q\)is true when both p and q are true and is false in all other cases.

The truth table for the conjunction\(p \wedge q\)is as follows:

\(p\,\,\)	\(\,q\)	\(p \wedge q\)
True	True	True
True	False	False
False	True	False
False	False	False

From the above table, observe that \(p \wedge q\)is true when both p and q are true.

Exclusion of propositions

Define the exclusion of the propositions\(p\,\,and\,\,q\) follows:

The exclusion of\(p\,\,and\,\,q\)can be denoted by\(p \oplus q\). The exclusion\(p \oplus q\)is true when exactly one of p and q is true and is false in all other cases.

The truth table for the exclusion\(p \oplus q\)is as follows:

\(p\,\,\)	\(\,q\)	\(p \oplus q\)
True	True	False
True	False	True
False	True	True
False	False	False

From the above table, observe that \(p \oplus q\)is true when exactly one of p and q is true.

Conditional statement of propositions

Define the conditional statement of the propositions\(p\,\,and\,\,q\)follows:

The conditional statement of\(p\,\,and\,\,q\)can be denoted by\(p \to q\), which can be read as “if p, then q”. The conditional statement\(p \to q\)is false when p is true and q is false, and is true in all other cases.

The truth table for the conditional statement\(p \to q\)is as follows:

\(p\,\,\)	\(\,q\)	\(p \to q\)
True	True	True
True	False	False
False	True	True
False	False	True

From the above table, observe that \(p \to q\)is false when p is true and q is false.

Bi-conditional statement of propositions

Define the bi-conditional statement of the propositions\(p\,\,and\,\,q\) follows:

The bi-conditional statement of\(p\,\,and\,\,q\)can be denoted by\(p \leftrightarrow q\), which can be read as “if p if and only if q”. The bi-conditional statement\(p \leftrightarrow q\)is true when p and q have the same truth values, and is false in all other cases.

The truth table for the bi-conditional statement\(p \leftrightarrow q\)is as follows:

\(p\,\,\)	\(\,q\)	\(p \to q\)
True	True	True
True	False	False
False	True	False
False	False	True

From the above table, observe that \(p \leftrightarrow q\)is true when p and q have the same truth values.

Consideration of statements

Let p be the statement “I’ll go to the movies tonight” and q is the statement “I’ll finish my discrete mathematics homework.”

Disjunction; conjunction; exclusion; conditional and bi-conditional statement of given proposition

Write the disjunction\(p \vee q\)as “I’ll go to the movies tonight or I’ll finish my discrete mathematics homework.”

Write the conjunction\(p \wedge q\)as I’ll go to the movies tonight and I’ll finish my discrete mathematics homework.”

Write the exclusion or\(p \oplus q\)as “I’ll go to the movies tonight or I’ll finish my discrete mathematics homework but not both.”

Write the conditional statement\(p \to q\)as “I’ll go to the movies tonight then I’ll finish my discrete mathematics homework.”

Write the bi-conditional statement\(p \leftrightarrow q\)as “I’ll go to the movies tonight if and only if I’ll finish my discrete mathematics homework.”